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A particle free to move along the \( X \)-axis has potential energy given as
\[
U(x)=k\left[1-\exp \left(-x^{2}\right)\right] \quad \text { (for }-\inftyx+\infty \text { ) }
\]
where, \( k \) is a positive constant of appropriate
dimensions. Then,
(a) at points away from origin, the particle is in equilibrium
(b) for any finite non-zero value of \( x \), there is a force directed away from the origin
(c) its total mechanical energy is \( k / 2 \) and it is equal to its kinetic energy at origin
(d) the motion of the particle is simple harmonic
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